Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
Require Import Coq.Init.Specif.
Definition judg2exprType (j:Judg) : Type :=
match j with
- (Γ > Δ > Σ |- τ) => forall (ξ:ExprVarResolver Γ) vars, Σ = mapOptionTree ξ vars ->
- FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ)
+ (Γ > Δ > Σ |- τ @ l) => forall (ξ:ExprVarResolver Γ) vars, Σ = mapOptionTree ξ vars ->
+ FreshM (ITree _ (fun t => Expr Γ Δ ξ t l) τ)
end.
- Definition justOne Γ Δ ξ τ : ITree _ (fun t => Expr Γ Δ ξ t) [τ] -> Expr Γ Δ ξ τ.
+ Definition justOne Γ Δ ξ τ l : ITree _ (fun t => Expr Γ Δ ξ t l) [τ] -> Expr Γ Δ ξ τ l.
intros.
inversion X; auto.
Defined.
Defined.
Lemma update_branches : forall Γ (ξ:VV -> LeveledHaskType Γ ★) lev l1 l2 q,
- update_ξ ξ lev (app l1 l2) q = update_ξ (update_ξ ξ lev l2) lev l1 q.
+ update_xi ξ lev (app l1 l2) q = update_xi (update_xi ξ lev l2) lev l1 q.
intros.
induction l1.
reflexivity.
Lemma fresh_lemma'' Γ
: forall types ξ lev,
FreshM { varstypes : _
- | mapOptionTree (update_ξ(Γ:=Γ) ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev)
+ | mapOptionTree (update_xi(Γ:=Γ) ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev)
/\ distinct (leaves (mapOptionTree (@fst _ _) varstypes)) }.
admit.
Defined.
Lemma fresh_lemma' Γ
: forall types vars Σ ξ lev, Σ = mapOptionTree ξ vars ->
FreshM { varstypes : _
- | mapOptionTree (update_ξ(Γ:=Γ) ξ lev (leaves varstypes)) vars = Σ
- /\ mapOptionTree (update_ξ ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev)
+ | mapOptionTree (update_xi(Γ:=Γ) ξ lev (leaves varstypes)) vars = Σ
+ /\ mapOptionTree (update_xi ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev)
/\ distinct (leaves (mapOptionTree (@fst _ _) varstypes)) }.
induction types.
intros; destruct a.
intros vars Σ ξ lev pf; refine (bind x2 = IHtypes2 vars Σ ξ lev pf; _).
apply FreshMon.
destruct x2 as [vt2 [pf21 [pf22 pfdist]]].
- refine (bind x1 = IHtypes1 (vars,,(mapOptionTree (@fst _ _) vt2)) (Σ,,(types2@@@lev)) (update_ξ ξ lev
+ refine (bind x1 = IHtypes1 (vars,,(mapOptionTree (@fst _ _) vt2)) (Σ,,(types2@@@lev)) (update_xi ξ lev
(leaves vt2)) _ _; return _).
apply FreshMon.
simpl.
Lemma fresh_lemma Γ ξ vars Σ Σ' lev
: Σ = mapOptionTree ξ vars ->
FreshM { vars' : _
- | mapOptionTree (update_ξ(Γ:=Γ) ξ lev ((vars',Σ')::nil)) vars = Σ
- /\ mapOptionTree (update_ξ ξ lev ((vars',Σ')::nil)) [vars'] = [Σ' @@ lev] }.
+ | mapOptionTree (update_xi(Γ:=Γ) ξ lev ((vars',Σ')::nil)) vars = Σ
+ /\ mapOptionTree (update_xi ξ lev ((vars',Σ')::nil)) [vars'] = [Σ' @@ lev] }.
intros.
set (fresh_lemma' Γ [Σ'] vars Σ ξ lev H) as q.
refine (q >>>= fun q' => return _).
inversion pf2.
Defined.
- Definition ujudg2exprType Γ (ξ:ExprVarResolver Γ)(Δ:CoercionEnv Γ) Σ τ : Type :=
- forall vars, Σ = mapOptionTree ξ vars -> FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ).
+ Definition ujudg2exprType Γ (ξ:ExprVarResolver Γ)(Δ:CoercionEnv Γ) Σ τ l : Type :=
+ forall vars, Σ = mapOptionTree ξ vars -> FreshM (ITree _ (fun t => Expr Γ Δ ξ t l) τ).
- Definition urule2expr : forall Γ Δ h j t (r:@Arrange _ h j) (ξ:VV -> LeveledHaskType Γ ★),
- ujudg2exprType Γ ξ Δ h t ->
- ujudg2exprType Γ ξ Δ j t
+ Definition urule2expr : forall Γ Δ h j t l (r:@Arrange _ h j) (ξ:VV -> LeveledHaskType Γ ★),
+ ujudg2exprType Γ ξ Δ h t l ->
+ ujudg2exprType Γ ξ Δ j t l
.
intros Γ Δ.
- refine (fix urule2expr h j t (r:@Arrange _ h j) ξ {struct r} :
- ujudg2exprType Γ ξ Δ h t ->
- ujudg2exprType Γ ξ Δ j t :=
+ refine (fix urule2expr h j t l (r:@Arrange _ h j) ξ {struct r} :
+ ujudg2exprType Γ ξ Δ h t l ->
+ ujudg2exprType Γ ξ Δ j t l :=
match r as R in Arrange H C return
- ujudg2exprType Γ ξ Δ H t ->
- ujudg2exprType Γ ξ Δ C t
+ ujudg2exprType Γ ξ Δ H t l ->
+ ujudg2exprType Γ ξ Δ C t l
with
- | RLeft h c ctx r => let case_RLeft := tt in (fun e => _) (urule2expr _ _ _ r)
- | RRight h c ctx r => let case_RRight := tt in (fun e => _) (urule2expr _ _ _ r)
- | RCanL a => let case_RCanL := tt in _
- | RCanR a => let case_RCanR := tt in _
- | RuCanL a => let case_RuCanL := tt in _
- | RuCanR a => let case_RuCanR := tt in _
- | RAssoc a b c => let case_RAssoc := tt in _
- | RCossa a b c => let case_RCossa := tt in _
- | RExch a b => let case_RExch := tt in _
- | RWeak a => let case_RWeak := tt in _
- | RCont a => let case_RCont := tt in _
- | RComp a b c f g => let case_RComp := tt in (fun e1 e2 => _) (urule2expr _ _ _ f) (urule2expr _ _ _ g)
+ | ALeft h c ctx r => let case_ALeft := tt in (fun e => _) (urule2expr _ _ _ _ r)
+ | ARight h c ctx r => let case_ARight := tt in (fun e => _) (urule2expr _ _ _ _ r)
+ | AId a => let case_AId := tt in _
+ | ACanL a => let case_ACanL := tt in _
+ | ACanR a => let case_ACanR := tt in _
+ | AuCanL a => let case_AuCanL := tt in _
+ | AuCanR a => let case_AuCanR := tt in _
+ | AAssoc a b c => let case_AAssoc := tt in _
+ | AuAssoc a b c => let case_AuAssoc := tt in _
+ | AExch a b => let case_AExch := tt in _
+ | AWeak a => let case_AWeak := tt in _
+ | ACont a => let case_ACont := tt in _
+ | AComp a b c f g => let case_AComp := tt in (fun e1 e2 => _) (urule2expr _ _ _ _ f) (urule2expr _ _ _ _ g)
end); clear urule2expr; intros.
- destruct case_RCanL.
+ destruct case_AId.
+ apply X.
+
+ destruct case_ACanL.
simpl; unfold ujudg2exprType; intros.
simpl in X.
apply (X ([],,vars)).
simpl; rewrite <- H; auto.
- destruct case_RCanR.
+ destruct case_ACanR.
simpl; unfold ujudg2exprType; intros.
simpl in X.
apply (X (vars,,[])).
simpl; rewrite <- H; auto.
- destruct case_RuCanL.
+ destruct case_AuCanL.
simpl; unfold ujudg2exprType; intros.
destruct vars; try destruct o; inversion H.
simpl in X.
apply (X vars2); auto.
- destruct case_RuCanR.
+ destruct case_AuCanR.
simpl; unfold ujudg2exprType; intros.
destruct vars; try destruct o; inversion H.
simpl in X.
apply (X vars1); auto.
- destruct case_RAssoc.
+ destruct case_AAssoc.
simpl; unfold ujudg2exprType; intros.
simpl in X.
destruct vars; try destruct o; inversion H.
apply (X (vars1_1,,(vars1_2,,vars2))).
subst; auto.
- destruct case_RCossa.
+ destruct case_AuAssoc.
simpl; unfold ujudg2exprType; intros.
simpl in X.
destruct vars; try destruct o; inversion H.
apply (X ((vars1,,vars2_1),,vars2_2)).
subst; auto.
- destruct case_RExch.
+ destruct case_AExch.
simpl; unfold ujudg2exprType ; intros.
simpl in X.
destruct vars; try destruct o; inversion H.
apply (X (vars2,,vars1)).
inversion H; subst; auto.
- destruct case_RWeak.
+ destruct case_AWeak.
simpl; unfold ujudg2exprType; intros.
simpl in X.
apply (X []).
auto.
- destruct case_RCont.
+ destruct case_ACont.
simpl; unfold ujudg2exprType ; intros.
simpl in X.
apply (X (vars,,vars)).
rewrite <- H.
auto.
- destruct case_RLeft.
+ destruct case_ALeft.
intro vars; unfold ujudg2exprType; intro H.
destruct vars; try destruct o; inversion H.
apply (fun q => e ξ q vars2 H2).
simpl.
reflexivity.
- destruct case_RRight.
+ destruct case_ARight.
intro vars; unfold ujudg2exprType; intro H.
destruct vars; try destruct o; inversion H.
apply (fun q => e ξ q vars1 H1).
simpl.
reflexivity.
- destruct case_RComp.
+ destruct case_AComp.
apply e2.
apply e1.
apply X.
Defined.
Definition letrec_helper Γ Δ l (varstypes:Tree ??(VV * HaskType Γ ★)) ξ' :
- ITree (LeveledHaskType Γ ★)
- (fun t : LeveledHaskType Γ ★ => Expr Γ Δ ξ' t)
- (mapOptionTree (ξ' ○ (@fst _ _)) varstypes)
+ ITree (HaskType Γ ★)
+ (fun t : HaskType Γ ★ => Expr Γ Δ ξ' t l)
+ (mapOptionTree (unlev ○ ξ' ○ (@fst _ _)) varstypes)
-> ELetRecBindings Γ Δ ξ' l varstypes.
intros.
induction varstypes.
simpl.
destruct (eqd_dec h0 l).
rewrite <- e0.
+ simpl in X.
+ subst.
apply X.
apply (Prelude_error "level mismatch; should never happen").
apply (Prelude_error "letrec type mismatch; should never happen").
exists x; auto.
Defined.
- Definition fix_indexing X (F:X->Type)(J:X->Type)(t:Tree ??{ x:X & F x })
- : ITree { x:X & F x } (fun x => J (projT1 x)) t
- -> ITree X (fun x:X => J x) (mapOptionTree (@projT1 _ _) t).
+ Definition fix_indexing X Y (J:X->Type)(t:Tree ??(X*Y))
+ : ITree (X * Y) (fun x => J (fst x)) t
+ -> ITree X (fun x:X => J x) (mapOptionTree (@fst _ _) t).
intro it.
induction it; simpl in *.
apply INone.
Defined.
Definition case_helper tc Γ Δ lev tbranches avars ξ :
- forall pcb:{sac : StrongAltCon & ProofCaseBranch tc Γ Δ lev tbranches avars sac},
- prod (judg2exprType (pcb_judg (projT2 pcb))) {vars' : Tree ??VV & pcb_freevars (projT2 pcb) = mapOptionTree ξ vars'} ->
+ forall pcb:(StrongAltCon * Tree ??(LeveledHaskType Γ ★)),
+ prod (judg2exprType (@pcb_judg tc Γ Δ lev tbranches avars (fst pcb) (snd pcb)))
+ {vars' : Tree ??VV & (snd pcb) = mapOptionTree ξ vars'} ->
((fun sac => FreshM
{ scb : StrongCaseBranchWithVVs VV eqdec_vv tc avars sac
- & Expr (sac_Γ sac Γ) (sac_Δ sac Γ avars (weakCK'' Δ)) (scbwv_ξ scb ξ lev) (weakLT' (tbranches @@ lev)) }) (projT1 pcb)).
+ & Expr (sac_gamma sac Γ) (sac_delta sac Γ avars (weakCK'' Δ)) (scbwv_xi scb ξ lev)
+ (weakT' tbranches) (weakL' lev) }) (fst pcb)).
intro pcb.
intro X.
simpl in X.
destruct s as [vars vars_pf].
refine (bind localvars = fresh_lemma' _ (unleaves (vec2list (sac_types sac _ avars))) vars
- (mapOptionTree weakLT' (pcb_freevars pcb)) (weakLT' ○ ξ) (weakL' lev) _ ; _).
+ (mapOptionTree weakLT' pcb) (weakLT' ○ ξ) (weakL' lev) _ ; _).
apply FreshMon.
rewrite vars_pf.
rewrite <- mapOptionTree_compose.
cut (distinct (vec2list localvars'')). intro H'''.
set (@Build_StrongCaseBranchWithVVs _ _ _ _ avars sac localvars'' H''') as scb.
- refine (bind q = (f (scbwv_ξ scb ξ lev) (vars,,(unleaves (vec2list (scbwv_exprvars scb)))) _) ; return _).
+ refine (bind q = (f (scbwv_xi scb ξ lev) (vars,,(unleaves (vec2list (scbwv_exprvars scb)))) _) ; return _).
apply FreshMon.
simpl.
- unfold scbwv_ξ.
+ unfold scbwv_xi.
rewrite vars_pf.
rewrite <- mapOptionTree_compose.
clear localvars_pf1.
Defined.
Definition gather_branch_variables
- Γ Δ (ξ:VV -> LeveledHaskType Γ ★) tc avars tbranches lev (alts:Tree ?? {sac : StrongAltCon &
- ProofCaseBranch tc Γ Δ lev tbranches avars sac})
+ Γ Δ
+ (ξ:VV -> LeveledHaskType Γ ★) tc avars tbranches lev
+ (alts:Tree ??(@StrongAltCon tc * Tree ??(LeveledHaskType Γ ★)))
:
forall vars,
- mapOptionTreeAndFlatten (fun x => pcb_freevars(Γ:=Γ) (projT2 x)) alts = mapOptionTree ξ vars
- -> ITree Judg judg2exprType (mapOptionTree (fun x => pcb_judg (projT2 x)) alts)
- -> ITree _ (fun q => prod (judg2exprType (pcb_judg (projT2 q)))
- { vars' : _ & pcb_freevars (projT2 q) = mapOptionTree ξ vars' })
+ mapOptionTreeAndFlatten (fun x => snd x) alts = mapOptionTree ξ vars
+ -> ITree Judg judg2exprType (mapOptionTree (fun x => @pcb_judg tc Γ Δ lev avars tbranches (fst x) (snd x)) alts)
+ -> ITree _ (fun q => prod (judg2exprType (@pcb_judg tc Γ Δ lev avars tbranches (fst q) (snd q)))
+ { vars' : _ & (snd q) = mapOptionTree ξ vars' })
alts.
induction alts;
intro vars;
simpl in *.
apply ileaf in source.
apply ILeaf.
- destruct s as [sac pcb].
+ destruct p as [sac pcb].
simpl in *.
split.
intros.
Defined.
+ Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★),
+ FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }.
+ intros Γ Σ.
+ induction Σ; intro ξ.
+ destruct a.
+ destruct l as [τ l].
+ set (fresh_lemma' Γ [τ] [] [] ξ l (refl_equal _)) as q.
+ refine (q >>>= fun q' => return _).
+ apply FreshMon.
+ clear q.
+ destruct q' as [varstypes [pf1 [pf2 distpf]]].
+ exists (mapOptionTree (@fst _ _) varstypes).
+ exists (update_xi ξ l (leaves varstypes)).
+ symmetry; auto.
+ refine (return _).
+ exists [].
+ exists ξ; auto.
+ refine (bind f1 = IHΣ1 ξ ; _).
+ apply FreshMon.
+ destruct f1 as [vars1 [ξ1 pf1]].
+ refine (bind f2 = IHΣ2 ξ1 ; _).
+ apply FreshMon.
+ destruct f2 as [vars2 [ξ2 pf22]].
+ refine (return _).
+ exists (vars1,,vars2).
+ exists ξ2.
+ simpl.
+ rewrite pf22.
+ rewrite pf1.
+ admit. (* freshness assumption *)
+ Defined.
+
+ Definition rlet Γ Δ Σ₁ Σ₂ σ₁ σ₂ p :
+ forall (X_ : ITree Judg judg2exprType
+ ([Γ > Δ > Σ₁ |- [σ₁] @ p],, [Γ > Δ > [σ₁ @@ p],, Σ₂ |- [σ₂] @ p])),
+ ITree Judg judg2exprType [Γ > Δ > Σ₁,, Σ₂ |- [σ₂] @ p].
+ intros.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+
+ refine (fresh_lemma _ ξ _ _ σ₁ p H2 >>>= (fun pf => _)).
+ apply FreshMon.
+
+ destruct pf as [ vnew [ pf1 pf2 ]].
+ set (update_xi ξ p (((vnew, σ₁ )) :: nil)) as ξ' in *.
+ inversion X_.
+ apply ileaf in X.
+ apply ileaf in X0.
+ simpl in *.
+
+ refine (X ξ vars1 _ >>>= fun X0' => _).
+ apply FreshMon.
+ simpl.
+ auto.
+
+ refine (X0 ξ' ([vnew],,vars2) _ >>>= fun X1' => _).
+ apply FreshMon.
+ simpl.
+ rewrite pf2.
+ rewrite pf1.
+ reflexivity.
+ apply FreshMon.
+
+ apply ILeaf.
+ apply ileaf in X1'.
+ apply ileaf in X0'.
+ simpl in *.
+ apply ELet with (ev:=vnew)(tv:=σ₁).
+ apply X0'.
+ apply X1'.
+ Defined.
+
+ Definition vartree Γ Δ Σ lev ξ :
+ forall vars, Σ @@@ lev = mapOptionTree ξ vars ->
+ ITree (HaskType Γ ★) (fun t : HaskType Γ ★ => Expr Γ Δ ξ t lev) Σ.
+ induction Σ; intros.
+ destruct a.
+ intros; simpl in *.
+ apply ILeaf.
+ destruct vars; try destruct o; inversion H.
+ set (EVar Γ Δ ξ v) as q.
+ rewrite <- H1 in q.
+ apply q.
+ intros.
+ apply INone.
+ intros.
+ destruct vars; try destruct o; inversion H.
+ apply IBranch.
+ eapply IHΣ1.
+ apply H1.
+ eapply IHΣ2.
+ apply H2.
+ Defined.
+
+
+ Definition rdrop Γ Δ Σ₁ Σ₁₂ a lev :
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- a,,Σ₁₂ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- a @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *.
+ intros.
+ set (X ξ vars H) as q.
+ simpl in q.
+ refine (q >>>= fun q' => return _).
+ apply FreshMon.
+ inversion q'.
+ apply X0.
+ Defined.
+
+ Definition rdrop' Γ Δ Σ₁ Σ₁₂ a lev :
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂,,a @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- a @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *.
+ intros.
+ set (X ξ vars H) as q.
+ simpl in q.
+ refine (q >>>= fun q' => return _).
+ apply FreshMon.
+ inversion q'.
+ auto.
+ Defined.
+
+ Definition rdrop'' Γ Δ Σ₁ Σ₁₂ lev :
+ ITree Judg judg2exprType [Γ > Δ > [],,Σ₁ |- Σ₁₂ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ eapply X with (vars:=[],,vars).
+ rewrite H; reflexivity.
+ Defined.
+
+ Definition rdrop''' Γ Δ a Σ₁ Σ₁₂ lev :
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > a,,Σ₁ |- Σ₁₂ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+ eapply X with (vars:=vars2).
+ auto.
+ Defined.
+
+ Definition rassoc Γ Δ Σ₁ a b c lev :
+ ITree Judg judg2exprType [Γ > Δ > ((a,,b),,c) |- Σ₁ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > (a,,(b,,c)) |- Σ₁ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+ destruct vars2; try destruct o; inversion H2.
+ apply X with (vars:=(vars1,,vars2_1),,vars2_2).
+ subst; reflexivity.
+ Defined.
+
+ Definition rassoc' Γ Δ Σ₁ a b c lev :
+ ITree Judg judg2exprType [Γ > Δ > (a,,(b,,c)) |- Σ₁ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > ((a,,b),,c) |- Σ₁ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+ destruct vars1; try destruct o; inversion H1.
+ apply X with (vars:=vars1_1,,(vars1_2,,vars2)).
+ subst; reflexivity.
+ Defined.
+
+ Definition swapr Γ Δ Σ₁ a b c lev :
+ ITree Judg judg2exprType [Γ > Δ > ((a,,b),,c) |- Σ₁ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > ((b,,a),,c) |- Σ₁ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+ destruct vars1; try destruct o; inversion H1.
+ apply X with (vars:=(vars1_2,,vars1_1),,vars2).
+ subst; reflexivity.
+ Defined.
+
+ Definition rdup Γ Δ Σ₁ a c lev :
+ ITree Judg judg2exprType [Γ > Δ > ((a,,a),,c) |- Σ₁ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > (a,,c) |- Σ₁ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+ apply X with (vars:=(vars1,,vars1),,vars2). (* is this allowed? *)
+ subst; reflexivity.
+ Defined.
+
+ (* holy cow this is ugly *)
+ Definition rcut Γ Δ Σ₃ lev Σ₁₂ :
+ forall Σ₁ Σ₂,
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > Σ₁₂ @@@ lev,,Σ₂ |- [Σ₃] @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > Σ₁,,Σ₂ |- [Σ₃] @ lev].
+
+ induction Σ₁₂.
+ intros.
+ destruct a.
+
+ eapply rlet.
+ apply IBranch.
+ apply X.
+ apply X0.
+
+ simpl in X0.
+ apply rdrop'' in X0.
+ apply rdrop'''.
+ apply X0.
+
+ intros.
+ simpl in X0.
+ apply rassoc in X0.
+ set (IHΣ₁₂1 _ _ (rdrop _ _ _ _ _ _ X) X0) as q.
+ set (IHΣ₁₂2 _ (Σ₁,,Σ₂) (rdrop' _ _ _ _ _ _ X)) as q'.
+ apply rassoc' in q.
+ apply swapr in q.
+ apply rassoc in q.
+ set (q' q) as q''.
+ apply rassoc' in q''.
+ apply rdup in q''.
+ apply q''.
+ Defined.
Definition rule2expr : forall h j (r:Rule h j), ITree _ judg2exprType h -> ITree _ judg2exprType j.
intros h j r.
refine (match r as R in Rule H C return ITree _ judg2exprType H -> ITree _ judg2exprType C with
- | RArrange a b c d e r => let case_RURule := tt in _
+ | RArrange a b c d e l r => let case_RURule := tt in _
| RNote Γ Δ Σ τ l n => let case_RNote := tt in _
| RLit Γ Δ l _ => let case_RLit := tt in _
| RVar Γ Δ σ p => let case_RVar := tt in _
| RAppCo Γ Δ Σ κ σ₁ σ₂ γ σ l => let case_RAppCo := tt in _
| RAbsCo Γ Δ Σ κ σ σ₁ σ₂ y => let case_RAbsCo := tt in _
| RApp Γ Δ Σ₁ Σ₂ tx te p => let case_RApp := tt in _
- | RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ p => let case_RLet := tt in _
- | RJoin Γ p lri m x q => let case_RJoin := tt in _
- | RVoid _ _ => let case_RVoid := tt in _
+ | RCut Γ Δ Σ Σ₁ Σ₁₂ Σ₂ Σ₃ l => let case_RCut := tt in _
+ | RLeft Γ Δ Σ₁ Σ₂ Σ l => let case_RLeft := tt in _
+ | RRight Γ Δ Σ₁ Σ₂ Σ l => let case_RRight := tt in _
+ | RVoid _ _ l => let case_RVoid := tt in _
| RBrak Σ a b c n m => let case_RBrak := tt in _
| REsc Σ a b c n m => let case_REsc := tt in _
| RCase Γ Δ lev tc Σ avars tbranches alts => let case_RCase := tt in _
destruct case_RURule.
apply ILeaf. simpl. intros.
- set (@urule2expr a b _ _ e r0 ξ) as q.
- set (fun z => q z) as q'.
- simpl in q'.
- apply q' with (vars:=vars).
- clear q' q.
+ set (@urule2expr a b _ _ e l r0 ξ) as q.
unfold ujudg2exprType.
+ unfold ujudg2exprType in q.
+ apply q with (vars:=vars).
intros.
apply X_ with (vars:=vars0).
auto.
destruct case_RVar.
apply ILeaf; simpl; intros; refine (return ILeaf _ _).
- destruct vars; simpl in H; inversion H; destruct o. inversion H1. rewrite H2.
- apply EVar.
+ destruct vars; simpl in H; inversion H; destruct o. inversion H1.
+ set (@EVar _ _ _ Δ ξ v) as q.
+ rewrite <- H2 in q.
+ simpl in q.
+ apply q.
inversion H.
destruct case_RGlobal.
refine (fresh_lemma _ ξ vars _ tx x H >>>= (fun pf => _)).
apply FreshMon.
destruct pf as [ vnew [ pf1 pf2 ]].
- set (update_ξ ξ x (((vnew, tx )) :: nil)) as ξ' in *.
+ set (update_xi ξ x (((vnew, tx )) :: nil)) as ξ' in *.
refine (X_ ξ' (vars,,[vnew]) _ >>>= _).
apply FreshMon.
simpl.
apply ileaf in X. simpl in X.
apply X.
- destruct case_RJoin.
- apply ILeaf; simpl; intros.
- inversion X_.
- apply ileaf in X.
- apply ileaf in X0.
- simpl in *.
- destruct vars; inversion H.
- destruct o; inversion H3.
- refine (X ξ vars1 H3 >>>= fun X' => X0 ξ vars2 H4 >>>= fun X0' => return _).
- apply FreshMon.
- apply FreshMon.
- apply IBranch; auto.
-
destruct case_RApp.
apply ILeaf.
inversion X_.
simpl in *.
apply (EApp _ _ _ _ _ _ q1' q2').
- destruct case_RLet.
- apply ILeaf.
- simpl in *; intros.
- destruct vars; try destruct o; inversion H.
- refine (fresh_lemma _ ξ vars1 _ σ₂ p H1 >>>= (fun pf => _)).
- apply FreshMon.
- destruct pf as [ vnew [ pf1 pf2 ]].
- set (update_ξ ξ p (((vnew, σ₂ )) :: nil)) as ξ' in *.
+ destruct case_RCut.
+ apply rassoc.
+ apply swapr.
+ apply rassoc'.
+
inversion X_.
- apply ileaf in X.
- apply ileaf in X0.
+ subst.
+ clear X_.
+
+ apply rassoc' in X0.
+ apply swapr in X0.
+ apply rassoc in X0.
+
+ induction Σ₃.
+ destruct a.
+ subst.
+ eapply rcut.
+ apply X.
+ apply X0.
+
+ apply ILeaf.
+ simpl.
+ intros.
+ refine (return _).
+ apply INone.
+ set (IHΣ₃1 (rdrop _ _ _ _ _ _ X0)) as q1.
+ set (IHΣ₃2 (rdrop' _ _ _ _ _ _ X0)) as q2.
+ apply ileaf in q1.
+ apply ileaf in q2.
simpl in *.
- refine (X ξ vars2 _ >>>= fun X0' => _).
+ apply ILeaf.
+ simpl.
+ intros.
+ refine (q1 _ _ H >>>= fun q1' => q2 _ _ H >>>= fun q2' => return _).
apply FreshMon.
- auto.
+ apply FreshMon.
+ apply IBranch; auto.
- refine (X0 ξ' (vars1,,[vnew]) _ >>>= fun X1' => _).
+ destruct case_RLeft.
+ apply ILeaf.
+ simpl; intros.
+ destruct vars; try destruct o; inversion H.
+ refine (X_ _ _ H2 >>>= fun X' => return _).
apply FreshMon.
- rewrite H1.
- simpl.
- rewrite pf2.
- rewrite pf1.
- rewrite H1.
- reflexivity.
+ apply IBranch.
+ eapply vartree.
+ apply H1.
+ apply X'.
- refine (return _).
+ destruct case_RRight.
apply ILeaf.
- apply ileaf in X0'.
- apply ileaf in X1'.
- simpl in *.
- apply ELet with (ev:=vnew)(tv:=σ₂).
- apply X0'.
- apply X1'.
+ simpl; intros.
+ destruct vars; try destruct o; inversion H.
+ refine (X_ _ _ H1 >>>= fun X' => return _).
+ apply FreshMon.
+ apply IBranch.
+ apply X'.
+ eapply vartree.
+ apply H2.
destruct case_RVoid.
apply ILeaf; simpl; intros.
apply ILeaf; simpl; intros.
refine (bind ξvars = fresh_lemma' _ y _ _ _ t H; _). apply FreshMon.
destruct ξvars as [ varstypes [ pf1[ pf2 pfdist]]].
- refine (X_ ((update_ξ ξ t (leaves varstypes)))
- (vars,,(mapOptionTree (@fst _ _) varstypes)) _ >>>= fun X => return _); clear X_. apply FreshMon.
+ refine (X_ ((update_xi ξ t (leaves varstypes)))
+ ((mapOptionTree (@fst _ _) varstypes),,vars) _ >>>= fun X => return _); clear X_. apply FreshMon.
simpl.
rewrite pf2.
rewrite pf1.
apply (@ELetRec _ _ _ _ _ _ _ varstypes).
auto.
apply (@letrec_helper Γ Δ t varstypes).
- rewrite <- pf2 in X1.
rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite pf2.
+ replace ((mapOptionTree unlev (y @@@ t))) with y.
+ apply X0.
+ clear pf1 X0 X1 pfdist pf2 vars varstypes.
+ induction y; try destruct a; auto.
+ rewrite IHy1 at 1.
+ rewrite IHy2 at 1.
+ reflexivity.
+ apply ileaf in X1.
+ simpl in X1.
apply X1.
- apply ileaf in X0.
- apply X0.
destruct case_RCase.
apply ILeaf; simpl; intros.
| scnd_branch _ _ _ c1 c2 => let case_branch := tt in fun q => IBranch _ _ (closed2expr _ _ c1 q) (closed2expr _ _ c2 q)
end.
- Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★),
- FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }.
- intros Γ Σ.
- induction Σ; intro ξ.
- destruct a.
- destruct l as [τ l].
- set (fresh_lemma' Γ [τ] [] [] ξ l (refl_equal _)) as q.
- refine (q >>>= fun q' => return _).
- apply FreshMon.
- clear q.
- destruct q' as [varstypes [pf1 [pf2 distpf]]].
- exists (mapOptionTree (@fst _ _) varstypes).
- exists (update_ξ ξ l (leaves varstypes)).
- symmetry; auto.
- refine (return _).
- exists [].
- exists ξ; auto.
- refine (bind f1 = IHΣ1 ξ ; _).
- apply FreshMon.
- destruct f1 as [vars1 [ξ1 pf1]].
- refine (bind f2 = IHΣ2 ξ1 ; _).
- apply FreshMon.
- destruct f2 as [vars2 [ξ2 pf22]].
- refine (return _).
- exists (vars1,,vars2).
- exists ξ2.
- simpl.
- rewrite pf22.
- rewrite pf1.
- admit.
- Defined.
-
- Definition proof2expr Γ Δ τ Σ (ξ0: VV -> LeveledHaskType Γ ★)
- {zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [τ]] ->
- FreshM (???{ ξ : _ & Expr Γ Δ ξ τ}).
+ Definition proof2expr Γ Δ τ l Σ (ξ0: VV -> LeveledHaskType Γ ★)
+ {zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [τ] @ l] ->
+ FreshM (???{ ξ : _ & Expr Γ Δ ξ τ l}).
intro pf.
set (mkSIND systemfc_all_rules_one_conclusion _ _ _ pf (scnd_weak [])) as cnd.
apply closed2expr in cnd.
auto.
refine (return OK _).
exists ξ.
- apply (ileaf it).
+ apply ileaf in it.
+ simpl in it.
+ apply it.
apply INone.
Defined.
projects / coq-hetmet.git / blobdiff